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The pattern of molecular evolution varies among gene sites and genes in a genome. By taking into account the complex heterogeneity of evolutionary processes among sites in a genome, Bayesian infinite mixture models of genomic evolution enable robust phylogenetic inference. With large modern data sets, however, the computational burden of Markov chain Monte Carlo sampling techniques becomes prohibitive. Here, we have developed a variational Bayesian procedure to speed up the widely used PhyloBayes MPI program, which deals with the heterogeneity of amino acid profiles. Rather than sampling from the posterior distribution, the procedure approximates the (unknown) posterior distribution using a manageable distribution called the variational distribution. The parameters in the variational distribution are estimated by minimizing Kullback-Leibler divergence. To examine performance, we analyzed three empirical data sets consisting of mitochondrial, plastid-encoded, and nuclear proteins. Our variational method accurately approximated the Bayesian inference of phylogenetic tree, mixture proportions, and the amino acid propensity of each component of the mixture while using orders of magnitude less computational time.

Stochastic Variational Inference for Bayesian Phylogenetics: A Case of CAT Model